Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C8×#731 ⋊ C2 |
| Sylow | #731 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#4184 |
| commutator, cosocle | C4×#731 | C22 |
| Frattini | C32×C4 | C2×C3⋊D3 |
| Fitting | C8×#731 | C2 |
| socle | C2×C3 | C4⋊#272 |
| hypercenter | C8 | #4184 |
| hypocenter | #731 | D8 |
| LCS | G ▷ C4×#731 ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
154 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^4,3^4|b^2=cd,c^2=d,f^3=h^2,b^a=bc,c^a=cd,e^a=e^2,f^a=f^2h,h^a=h^2,f^e=fg,g^e=gh,g^f=gh^277 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 470);